Dimension growth for iterated sumsets

TL;DR

This paper investigates the growth of dimensions in sumsets and iterated sumsets, establishing conditions under which these dimensions increase significantly, with applications to fractal sets and distance problems.

Contribution

It introduces natural conditions ensuring dimension growth in sumsets, leveraging Hochman's inverse entropy theorem and analyzing Assouad and lower dimensions.

Findings

Sumsets can have strictly larger upper box dimension than the original set.

Iterated sumsets can approach full dimension under certain conditions.

Results apply to uniformly perfect and Ahlfors-David regular sets.

Abstract

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$. Our results apply to, for example, all uniformly perfect sets, which include Ahlfors-David regular sets. Our proofs rely on Hochman's inverse theorem for entropy and the Assouad and lower dimensions play a critical role. We give several applications of our results including an Erd\H{o}s-Volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension.